\[ \int \frac {1}{\sqrt {-x} \sqrt {a-b x} \sqrt {a+b x}} \, dx \]
Optimal antiderivative \[ -\frac {2 \EllipticF \left (\frac {\sqrt {b}\, \sqrt {-x}}{\sqrt {a}}, i\right ) \sqrt {a}\, \sqrt {1-\frac {b x}{a}}\, \sqrt {1+\frac {b x}{a}}}{\sqrt {b}\, \sqrt {-b x +a}\, \sqrt {b x +a}} \]
command
integrate(1/(-x)^(1/2)/(-b*x+a)^(1/2)/(b*x+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )}{b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b x + a} \sqrt {-b x + a} \sqrt {-x}}{b^{2} x^{3} - a^{2} x}, x\right ) \]